Quadratic points on bielliptic modular curves

Abstract

Bruin and Najman [6], Ozman and Siksek [28], and Box [4] described all the quadratic points on the modular curves of genus 2 ≤ g(X 0 (n)) ≤ 5. Since all the hyperelliptic curves X 0 (n) are of genus ≤ 5 and as a curve can have infinitely many quadratic points only if it is either of genus ≤ 1, hyperelliptic or bielliptic, the question of describing the quadratic points on the bielliptic modular curves X 0 (n) naturally arises; this question has recently also been posed by Mazur.We answer Mazur's question completely and describe the quadratic points on all the bielliptic modular curves X 0 (n) for which this has not been done already. The values of n that we deal with are n = 60, 62, 69, 79, 83, 89, 92, 94, 95, 101, 119 and 131; the curves X 0 (n) are of genus up to 11. We find all the exceptional points on these curves and show that they all correspond to CM elliptic curves. The two main methods we use are Box's relative symmetric Chabauty method and an application of a moduli description of Q-curves of degree d with an independent isogeny of degree m, which reduces the problem to finding the rational points on several quotients of modular curves.cases in detail in Section 7.1.The code that verifies all our computations can be found on: https://github.com/brutalni-vux/QuadPtsBielliptic.

On elliptic curves with p-isogenies over quadratic fields

“…For p ≤ 73 the papers [7] and [6] compute all quadratic points x ∈ X 0 (p)(K) that satisfy w p (x) x τ ; such points are called exceptional. The recent paper [18] does the same for p ≥ 79, and we will use this result for p = 101 in our example in Section 5.3. In each paper, we simply consult the tables and read off the possible fields over which these exceptional quadratic points are defined.…”

Cyclic isogenies of elliptic curves over a fixed quadratic field

“…Catalogues of quadratic points. We use existing classifications of quadratic points on low-genus modular curves, due to several independent works in recent years; in order of appearance, these are: Bruin-Najman [8], Özman-Siksek [29], Box [7], Najman-Vukorepa [27], and Vukorepa [34]. In short, quadratic points on X 0 (N) are classified non-exceptional or exceptional according to whether or not they arise as pullbacks of Q-rational points on a quotient X 0 (N)/w d (d is usually chosen such that the quotient is of minimal genus).…”

“…Although the subject of determining rational points on modular curves has seen much recent development (see e.g. [7,1,27]) , some of these composite values for quadratic fields with small absolute value of discriminant are beyond current methods. For this reason, we search for 'convenient' quadratic fields K for which (among other conditions) the largest value in S(K) is 169.…”

On elliptic curves with $p$-isogenies over quadratic fields

“…For p ≤ 73 the papers [6] and [5] compute all quadratic points x ∈ X 0 (p)(K) that satisfy w p (x) = x τ ; such points are called exceptional. The recent paper [21] does the same for p ≥ 79, and we will use this for p = 79 in Theorem 5.2 below, and for p = 101 in our example in Section 5.3. In each paper, we simply consult the tables and read off the possible fields over which these exceptional quadratic points are defined.…”

“…The genera of the curves X 0 (97), X 0 (103), and X 0 (157) are 7, 8, and 12, with Atkin-Lehner quotients X 0 (p)/w p of genera 3, 2, and 5 respectively. We believe it should be possible to deal with the cases 97 and 103, and possibly 157 (although perhaps the genera are too large here), using the techniques of [5,18,21]. We hope to pursue this in future work.…”